I am being asked to define different subgroups of the general linear group as stabilizers of group actions on matrix spaces. As of now I have found these :
The orthogonal group as a stabilizer of a quadratic form for the congruence action.
The symplectic group as a stabiliser of a symplectic form for a similar action.
The upper triangular matrices group as a stabilizer of a complete flag of vector subspaces for it's natural action.
In this question, they talk about a decompostion with direct sums, but I am not sure : what subgroup would that bring?
Do you have other examples of subgroups of GL(E) realised with group actions? (Either by a stabilizer or by other means).
I would be curious to find out about your examples. Perhaps representation theory can help here.